Since Harris [1]} first proposed his conjecture on the question of central limit theorems for a branching random walk, the topic has been widely studied and in various forms. See e.g. [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]}, [14]}, [15]}, [16]} .
